Since \(U_{1}U_{2}=U\)  and \(U_{2}U_{1}=U_{2}UU_{2}^{-1}\)  are related
by a similarity transformation, they describe the same parent Floquet
system up to a half-period shift of the initial evolution time. The
system described by \(U_{1/2}^{2}\)  can thus be viewed as two equivalent
copies of \(U\)  up to a global phase shift \(\pi \) . Therefore, \(U_{1/2}^{2}\) 
and \(U\)  are expected to share the same topological features concerning
the stroboscopic dynamics, and we could view \(U_{1/2}\)  as a nontrivial
square-root of \(U\)  in the spirit of Dirac's taking square-root to
reach his equation for electrons [1]}. The dynamical and topological properties
of \(U\)  can further be carried over to \(U_{1/2}\) , which are first confirmed
by explicit studies of Floquet topological superconductors and time
crystals in Ref. [2]}.
